Such value-prices may be described as ‘surplus-levelling’. They are themselves members of a much larger tribe which might be described as ‘bi-polar’ in that they
offer a twofold explanation of value, whereby the main element of it is traced to a set of enumerated production factors exerting their power individually by ‘absorp-
tion’, and a secondary constituent, the ‘surplus’, ascribed to all factors collaborat- ing conjointly in creating a value-element in addition to these. To the extent that
production factors may be defined as the only sectoral unit cost-indicators dis- played in the matrix W which can account for differential values pW in the
production sectors, any value-element remaining over and above these p − pW can only be explained as an undifferentiated and therefore uniform force acting in
supplementation of them, i.e. p − pW = kpW. This immediately establishes the value-prices p as the elements of a left-hand eigenvector of the matrix W.
4. The general n-sector case
The starting point is the ‘generator matrix’ a square matrix of order n the number of sectors such as
W = Á
à Ã
à Ä
a
1
b
1
..... n
1
a
2
b
2
..... n
2
. .
..... .
a
n
b
n
..... n
n
 Ã
à Ã
Å where the elements a, b, etc., may be termed the ‘generators’, being, as it were, the
seed-corn of the left-hand eigenvector of W. The generator in position ij of W measures the physical input of product i or any other cost item involving the use of
i per unit output i.
3
The left-hand eigenvector of such a matrix, say p, is defined by
the equation pw = pW and will therefore satisfy the equation pW − wI
pW
0 =0 1
where w is the dominant eigenvalue of W which makes the characteristic determi- nant
W−wI=W0 vanish. It follows that p will be proportional to the set of first-column cofactors of the characteristic matrix W
0 , say A=[A
1
, A
2
, …, A
n
], since Aa is the expansion of
W0 by its first column and must therefore vanish, while at the same time the product of A with all other columns of W
0 must also vanish since they are the expansions of
W0 by ‘alien’ cofactors. The same holds of all other columns of W
0 , since by virtue of their mutual proportionality, ensured by
3
Throughout this paper the matrix W is assumed to be strictly positive and non singular. Nor is this
implausible; for there is nothing to compel us to isolate costless goods, if any, in special sectors. They can always be merged with normal productions into composite sectors within a scheme which is of
necessity aggregative. I am indebted to Professor A. Bro´dy for this important caveat.
their derivation from a singular matrix, they will all give the same proportional results as Aa, thus all obeying Eq. 1 which establishes proportionality with p.
This fact may be expressed more concisely by the equation
p = ki
r
W 2
implying relativities p
1
p
2
= A
r
A
s
, where k is a proportionality factor, i
r
, the rth row of the unit vector, and r may be any number from 1 to n. Eq. 2 enables us
to convert the market prices of all commodities and of any indicator or aggregate derived from them into the equivalent measure in value prices, but does so only up
to a proportionality factor. To establish an absolute conversion we must in addition ‘scale’ the elements of p to ensure the invariance of any chosen indicator or
aggregate in the face of this conversion. The most convenient aggregate from this point of view would seem to be the national income py, where y is the column
vector of deliveries to final purposes, requiring k of Eq. 2 to be derived from the supplementary equation
ki
r
W
0 y=y 3
This ensures that the weighted average of the conversion factors in p is equal to unity, and therefore any element of it greatersmaller than unity can justifiably be
interpreted as the undervaluationovervaluation of the corresponding commodity by the market.
5. Technological change
